What's the problem with Framton Nuttel in "The Open Window"?
The answer to your question can be found in the beginning of the short story, where Framton and Vera meet for the first time and, already, the young man begins to feel uneasy about having come to visit at all. This is because he has a nervous condition from which he is hoping to fully recover by visiting the countryside.
Privately he doubted more than ever whether these formal visits on a succession of total strangers would do much towards helping the nerve cure which he was supposed to be undergoing.
Regarding his nervous condition, Framton Nuttel also offers the recommendation of his doctors.
...complete rest, an absence of mental excitement, and avoidance of anything in the nature of violent physical exercise...
His problem is, therefore, that he has been trying to find ways to get his "nerve cure" by visiting "a succession of total strangers." This could do very little to improve his condition, as one is never completely comfortable when surrounded by unknown people. If anything, it may make his condition worse.
There is more. The fact that his sister has to write recommendation letters on his behalf, to old friends, shows that Framton is also socially awkward and easily traumatized. This latter description fits well, considering that all it took was Vera's story about the open window for him to not just completely believe it, but also to completely lose control of himself and leave the house in horror.
Therefore, aside from his odd behaviors, Framton is unsuccessfully trying to get a cure for his condition, without a lot of success.
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